RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES

作者：Tao Terence^*; Vu Van

来源：Annals of Probability, 2012, 40(3): 1285-1315.

DOI：10.1214/11-AOP648

• Four moment theorem
• universality
• covariance matrices

We study the eigenvalues of the covariance matrix 1/n M*M of a large rectangular matrix M = M-n,M-p = (zeta(ij))(1 <= i <= p;1 <= j <= n) whose entries are i.i.d. random variables of mean zero, variance one, and having finite C(0)th moment for some sufficiently large constant C-0.
The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549-572])). We can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions.
As a byproduct of our arguments, we also extend our previous results on random Hermitian matrices to the case in which the entries have finite C(0)th moment rather than exponential decay.

• 出版日期2012-5